Fractional diffusion equation-based image denoising model using CN–GL scheme

References

• M. Abbaszadeh and M. Dehghan, A meshless numerical procedure for solving fractional reaction sub diffusion model via a new combination of alternating direction implicit (ADI) approach and interpolating element free Galerkin (EFG) method, Comput. Math. Appl. 70 (2015), pp. 2493–2512. doi: 10.1016/j.camwa.2015.09.011
   Web of Science ®Google Scholar
• M.J. Black, G. Sapiro, D.H. Marimont, and D. Heeger, Robust anisotropic diffusion, IEEE. Trans. Image Process. 7(3) (1998), pp. 421–432. doi: 10.1109/83.661192
   PubMed Web of Science ®Google Scholar
• F. Catte, P.-L. Lions, J.-M. Morel, and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM. J. Numer. Anal. 29(1) (1992), pp. 182–193. doi: 10.1137/0729012
   Web of Science ®Google Scholar
• Q. Chang and Z. Huang, Efficient algebraic multigrid algorithms and their convergence, SIAM J. Sci. Comput. 24 (2004), pp. 597–618. doi: 10.1137/S1064827501389850
   Web of Science ®Google Scholar
• K. Chen, Adaptive smoothing via contextual and local discontinuities, IEEE Trans. Pattern Anal. Mach. Intell. 27(10) (2005), pp. 1552–1567. doi: 10.1109/TPAMI.2005.190
   PubMed Web of Science ®Google Scholar
• K. Chen, Introduction to variational image-processing models and applications, Int. J. Comput. Math. 90 (2013), pp. 1–8. doi: 10.1080/00207160.2012.757073
   Web of Science ®Google Scholar
• M.H. Chen and W.H. Deng, A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation, Appl. Math. Model. 38 (2014), pp. 3244–3259. doi: 10.1016/j.apm.2013.11.043
   Web of Science ®Google Scholar
• E. Cuesta, M. Kirane, and S.A. Malik, Image structure preserving denoising using generalized fractional time integrals, Signal Process. 92 (2012), pp. 553–563. doi: 10.1016/j.sigpro.2011.09.001
   Web of Science ®Google Scholar
• M. Dehghan, A new ADI technique for two-dimensional parabolic equation with an integral condition, Comput. Math. Appl. 43 (2002), pp. 1477–1488. doi: 10.1016/S0898-1221(02)00113-X
   Web of Science ®Google Scholar
• M. Dehghan, M. Abbaszadeh, and A. Mohebbi, Analysis of a meshless method for the time fractional diffusion-wave equation, Commun. Nonlinear Sci. Numer. Simul. 73 (2016), pp. 445–476.
   Google Scholar
• D. Firsov and S.H. Lui, Domain decomposition methods in image denoising using Gaussian curvature, J. Comput. Appl. Math. 193(2) (2006), pp. 460–473. doi: 10.1016/j.cam.2005.05.032
   Web of Science ®Google Scholar
• B. Ghanbari, L. Rada, and K. Chen, A restarted iterative homotopy analysis method for two nonlinear models from image processing, Int. J. Comput. Math. 91 (2014), pp. 661–687. doi: 10.1080/00207160.2013.807340
   Web of Science ®Google Scholar
• Z. Guo, J. Sun, D. Zhang, and B. Wu, Adaptive Perona–Malik model based on the variable exponent for image denoising, IEEE. Trans. Image Process. 21(3) (2012), pp. 958–967. doi: 10.1109/TIP.2011.2169272
   PubMed Web of Science ®Google Scholar
• M.R. Hajiaboli, M.O. Ahmad, and C. Wang, An edge-adapting Laplacian kernel for nonlinear diffusion filters, IEEE. Trans. Image Process. 21(4) (2012), pp. 1561–1572. doi: 10.1109/TIP.2011.2172803
   PubMed Web of Science ®Google Scholar
• J. Hu, Y. Pu, and J. Zhou, A novel image denoising algorithm based on Riemann–Liouville definition, J. Comput. 6 (2011), pp. 1332–1338.
   Google Scholar
• M. Kamranian, M. Dehghan, and M. Tatari, An image denoising approach based on a meshfree method and the domain decomposition technique, Eng. Anal. Bound. Elem. 39 (2014), pp. 101–110. doi: 10.1016/j.enganabound.2013.11.003
   Web of Science ®Google Scholar
• C. Ke, W. Feng, and S. Qing-Wei, A novel image denoising algorithm based on Crank–Nicholson semi-implicit difference scheme, Prodedia Eng. 23 (2011), pp. 647–652. doi: 10.1016/j.proeng.2011.11.2560
   Google Scholar
• S. Kim, Partial differential equation-based image restoration: A hybrid model and color image denoising, IEEE. Trans. Image. Process. 15 (2006), pp. 1163–1170. doi: 10.1109/TIP.2005.864184
   PubMed Web of Science ®Google Scholar
• K. Krissian, C.-F. Westin, R. Kikinis, and K.G. Vosburgh, Oriented speckle reducing anisotropic diffusion, IEEE. Trans. Image. Process. 16(5) (2007), pp. 1412–1424. doi: 10.1109/TIP.2007.891803
   PubMed Web of Science ®Google Scholar
• G. Liu, T.-Z. Huang, and J. Liu, High-order TV-L1 based images restoration and spatially adapted regularization parameter selection, Comput. Math. Appl. 67 (2014), pp. 2015–2026. doi: 10.1016/j.camwa.2014.04.008
   Web of Science ®Google Scholar
• B. Mathieu, P. Melchior, A. Oustaloup, and C. Ceyral, Fractional differentiation for edge detection, Signal Process. 83 (2003), pp. 2421–2432. doi: 10.1016/S0165-1684(03)00194-4
   Web of Science ®Google Scholar
• M.M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection dispersion flow equations, J. Comput. Appl. Math. 172 (2003), pp. 65–77. doi: 10.1016/j.cam.2004.01.033
   Web of Science ®Google Scholar
• M.M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006), pp. 80–90. doi: 10.1016/j.apnum.2005.02.008
   Web of Science ®Google Scholar
• M.K. Ng, R.H. Chan, and W.-C. Tang, A Fast algorithm for deblurring models with Neumann boundary conditions, SIAM. J. Sci. Comput. 21 (1999), pp. 851–866. doi: 10.1137/S1064827598341384
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
   Google Scholar
• H.K. Rafsanjani, M.H. Sedaaghi, and S. Saryazdi, Efficient diffusion coefficient for image denoising, Comput. Math. Appl. 72 (2016), pp. 893–903. doi: 10.1016/j.camwa.2016.06.005
   Web of Science ®Google Scholar
• R.D. Richtmyer and K.W. Morton, Difference Methods for Initial Value Problems, 2nd ed., Krieger, Malabar, FL, 1994.
   Google Scholar
• L.I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), pp. 259–268. doi: 10.1016/0167-2789(92)90242-F
   Web of Science ®Google Scholar
• A. Saadatmandi, M. Dehghan, and M.-R. Azizi, The Sinc–Legendre collocation method for a class of fractional convection diffusion equations with variable coefficients, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), pp. 4125–4136. doi: 10.1016/j.cnsns.2012.03.003
   Web of Science ®Google Scholar
• G. Sudha Priya, P. Prakash, J.J. Nieto, and Z. Kayar, Higher-order numerical scheme for the fractional heat equation with Dirichlet and Neumann boundary conditions, Numer. Heat Transf. B 63 (2013), pp. 540–559. doi: 10.1080/10407790.2013.778719
   Web of Science ®Google Scholar
• B.J. Szekeres and F. Lzsák, Finite difference method for fractional diffusion equations with Neumann boundary conditions, Open Math. 13 (2015), pp. 581–600. doi: 10.1515/math-2015-0056
   Web of Science ®Google Scholar
• C. Tadjeran and M.M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys. 220 (2007), pp. 813–823. doi: 10.1016/j.jcp.2006.05.030
   Web of Science ®Google Scholar
• K. Thangavel, R. Manavalan, and I.L. Aroquiaraj, Removal of speckle noise from ultrasound medical image based on special filters-comparing study, J. Graph Vision Image Process. 9(3) (2009), pp. 25–32.
   Google Scholar
• W.Y. Tian, H. Zhou, and W. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp. 84 (2015), pp. 1703–1727. doi: 10.1090/S0025-5718-2015-02917-2
   Web of Science ®Google Scholar
• Y.L. You, W. Xu, A. Tannenbaum, and M. Kaveh, Behavioral analysis of anisotropic diffusion in image processing, IEEE. Trans. Image Process. 5(11) (1996), pp. 1539–1553. doi: 10.1109/83.541424
   PubMed Web of Science ®Google Scholar